MATH 256 Section 202
W2
Midterm Exam 2
March 15, 2017
Time Limit: 45 Minutes
Last Name:
First Name:
Student #:
This exam contains 11 pages (including this cover page) and 3 problems. Enter all requested information on the top of this page.
The following rules apply:
• You may not use your books, notes, or any calculator on this exam.
Problem
Points
• Unless a question asks you to state, or write down
the answer, you must show all your working.
There is credit given for using the correct method
to solve each problem. Unsupported final answers will not receive full credit.
1
6
2
16
3
16
• Organize your work, in a reasonably neat and
coherent way, in the space provided. Work scattered all over the page without a clear ordering is
very difficult to mark and might receive reduced
credit if your argument is not clear.
Total:
38
• If you need more space, use the back of the pages.
Do not write in the table to the right.
Do not open the exam until instructed to do so.
Score
MATH 256 Section 202
Midterm Exam 2 - Page 2 of 11
March 15, 2017
1. (6 points) Consider the following differential equation for y(x), where ω is some real number:
y 00 + ω 2 y = 0.
(a) (4 points) Show that if we impose the boundary conditions y(0) = 0 and y(1) = 0 then
there is a non-zero solution (i.e. it is not true that y(x) = 0 for all x) only if ω = nπ for
some non-zero integer n = . . . , −2, −1, 1, 2, . . . :
MATH 256 Section 202
Midterm Exam 2 - Page 3 of 11
March 15, 2017
(b) (2 points) Now consider the differential equation
y 00 + ky 0 + π 2 y = cos(πx),
where k > 0 is a positive real number. For what value of k is the differential equation
critically damped?
MATH 256 Section 202
Midterm Exam 2 - Page 4 of 11
March 15, 2017
2. (16 points) (a) (3 points) Consider the following figure which shows the direction field for a
homogeneous 2-dimensional system of ordinary differential equations. The magitude of the
eigenvalues are given by 7 and 3. The eigenvectors are shown by the solid straight lines.
What is the general solution of the system? You need to use the figure to find the eigenvectors, determine the sign of each eigenvalue, and decide which eigenvalue corresponds
with which eigenvector.
MATH 256 Section 202
Midterm Exam 2 - Page 5 of 11
March 15, 2017
(b) (3 points) Consider the following system of linear first order ODEs, where a is a real
number.
1 −1 − 3a
ẏ =
y.
1 −1 − 2a
Show that the system represents a stable spiral if 0 < a < 1.
MATH 256 Section 202
Midterm Exam 2 - Page 6 of 11
March 15, 2017
(c) (10 points) Find the solution to the system in (b) when a = 1, subject to the initial
condition
1
y(0) =
.
1
MATH 256 Section 202
More space for 2(b):
Midterm Exam 2 - Page 7 of 11
March 15, 2017
MATH 256 Section 202
Midterm Exam 2 - Page 8 of 11
March 15, 2017
3. (16 points) (a) (3 points) Consider the following figure showing the function g(t), which consists of a number of straight lines. Write down an expression for g(t) using the Heaviside
step function H(t). You may assume that g(t) = 0 for t ≤ 0 and t ≥ 5.
MATH 256 Section 202
Midterm Exam 2 - Page 9 of 11
March 15, 2017
(b) (4 points) Calculate the Laplace transform F (s) of the following function f (t) explicitly,
using the integral formula. State the range of s for which F (s) exists.


0 for t < 0
f (t) = t for 0 ≤ t < 1

 −(t−1)
e
for t ≥ 1
MATH 256 Section 202
Midterm Exam 2 - Page 10 of 11
March 15, 2017
(c) (5 points) The Laplace transform Y (s) of the solution y(t) to a linear second-order differential equation is given by
e−3s
Y (s) = 2
.
s + 3s + 2
Find y(t). You may use the results contained in the following table of Laplace transforms
in which G(s) is the Laplace transform of g(t), the Heaviside step function is written H(t),
and δ(t) is the Dirac δ-function, and a, c and ω are some constants.
f (t)
1
tn
sin(ωt)
cos(ωt)
δ(t − c)
ẏ
ÿ
eat g(t)
H(t − c)g(t − c)
F (s)
1/s
n!/sn+1
ω/(s2 + ω 2 )
s/(s2 + ω 2 )
e−cs
sY (s) − y(0)
2
s Y (s) − ẏ(0) − sy(0)
G(s − a)
e−sc G(s)
MATH 256 Section 202
Midterm Exam 2 - Page 11 of 11
March 15, 2017
More space for 3(c):
(d) (4 points) Write down a second-order differential equation and initial conditions for y(t)
for which the Laplace transform Y (s) of y(t) would be given by the expression for Y (s) in
3(c). You may use the results contained in the table of Laplace transforms on the previous
page.